Permanence for Nicholson-type Delay Systems with Patch Structure and Nonlinear Density-dependent Mortality Terms*

In this paper, we study the Nicholson-type delay systems with patch structure and nonlinear density-dependent mortality terms. Under appropriate conditions, we establish some criteria to ensure the permanence of this model. Moreover, we give some examples to illustrate our main results.


Introduction
To reveal the rule of population of the Australian sheep blowfly that obtained in experimental data [1], Gurney et al [2] put forward the following Nicholson's blowflies model N ′ (t) = −δN (t) + pN (t − τ )e −aN (t−τ ) . (1.1) Here, N (t) is the size of the population at time t, p is the maximum per capita daily egg production, 1 a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time.As a class of biological systems, *This work was supported by National Natural Science Foundation of China (grant nos.11101283, 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of P.R.China (grant no.LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R.China (grant no.Z201122436).† Corr.author.Tel.:+86 02136430738; fax: +86 02136430738.E-mail:chenweiwang2009@yahoo.com.cnEJQTDE, 2012 No. 73, p. 1 Nicholson's blowflies model and its analogous equation have attracted much attention.There have been a large number of results about this model and its modifications.We refer the reader to [3][4][5][6][7][8][9] and the references cited therein.Moreover, the main focus of Nicholson's blowflies model is on the scalar equation and results about patch structure of this model are gained rarely (see e.g.[10][11][12][13] and the reference therein).On the other hand, L. Berezansky et al [9] pointed out that a new study indicates that a linear model of density-dependent mortality will be most accurate for populations at low densities and marine ecologists are currently in the process of constructing new fishery models with nonlinear density-dependent mortality rates.Consequently, B. Liu and S. Gong [14] and Liu [15] presented extensive results on the permanence of the following Nicholson's blowflies model with a nonlinear density-dependent mortality term where P is a positive constant and function D might have one of the following forms: However, to the best of our knowledge, there have been few publications concerned with the permanence for Nicholson-type delay system with patch structure and nonlinear densitydependent mortality terms.Motivated by this, the main purpose of this paper is to give the conditions to guarantee the permanence for the following Nicholson-type delay system with patch structure and nonlinear density-dependent mortality terms: where a ij , b ij , c ik , γ ik : R → (0, +∞) are all continuous functions bounded above and below by positive constants, and τ ik (t) ≥ 0 are bounded continuous functions, we assume that a ij (t) > b ij (t) for t ∈ R and i, j = 1, 2 • • • , n, which show the biological significance of the mortality terms.
For convenience, we introduce some notations.Throughout this paper, given a bounded continuous function g defined on R, let g + and g − be defined as EJQTDE, 2012 No. 73, p. 2 Let R n (R n + ) be the set of all (nonnegative) real vectors, we will use x = (x 1 , . . ., x n ) T ∈ R n to denote a column vector, in which the symbol () T denotes the transpose of a vector.we let |x| denote the absolute-value vector given by |x| + ) as Banach space equipped with the supremum norm defined by ||ϕ|| = sup ) with t 0 , ν ∈ R 1 and i = 1, . . ., n, then we define x t ∈ C as x t = (x 1 t , . . .x n t ) T where x i t (θ) = x i (t + θ) for all θ ∈ [−r i , 0] and i = 1, . . ., n.
The initial conditions associated with system (1.3) are of the form: (1.4) We write N t (t 0 , ϕ)(N (t; t 0 , ϕ)) for a solution of the initial value problem (1.3) and (1.4) .
Definition 1.1.The system (1.3) with initial conditions (1.4) is said to be permanent, if there are positive constants k i and K i such that The remaining part of this paper is organized as follows.In sections 2 and 3, we shall derive new sufficient conditions for checking the permanence of model (1.3).In Section 4, we shall give some examples and remarks to illustrate our results obtained in the previous sections.
So there exist positive constants L i such that In what follows, we prove that there exists a positive constant l i such that Assume that (3.7) does not hold, then it exists For each t ≥ t 0 , we define Letting t → +∞, (3.9) implies that which contradicts with the inequality of (3.1).This ends the proof of Theorem 3.1.

Some examples
In this section we present some examples to illustrate our results.
We have  which is a contradiction.This implies that (4.5) holds and the system (4.3) with initial condition ϕ * is not permanent but extinct.
Remark 4.1.To the best of our knowledge, few authors have considered the problems of the permanence of Nicholson's blowflies model with patch structure and nonlinear densitydependent mortality terms.It is clear that all the results in [12][13][14][15] and the references therein cannot be applicable to prove the permanence of (4.1) and (4.2).This implies that the results of this paper are new.

2 j=1c≤
ij (t 2 )e λr i e + 4e λ )e < 0, , • • • , n. Letting n → +∞, from (2.12)-(2.14)we know that sup t∈R The above two examples that satisfy the conditions of Theorem 2.1 and Theorem 3.1 respectively are permanent.Next we shall give the example that does not satisfy the conditions of Theorem 2.1 is not permanent.